.TH std::assoc_legendre,std::assoc_legendref,std::assoc_legendrel 3 "2024.06.10" "http://cppreference.com" "C++ Standard Libary"
.SH NAME
std::assoc_legendre,std::assoc_legendref,std::assoc_legendrel \- std::assoc_legendre,std::assoc_legendref,std::assoc_legendrel

.SH Synopsis
   double      assoc_legendre( unsigned int n, unsigned int m, double x );

   double      assoc_legendre( unsigned int n, unsigned int m, float x );
   double      assoc_legendre( unsigned int n, unsigned int m, long double x );  \fB(1)\fP
   float       assoc_legendref( unsigned int n, unsigned int m, float x );

   long double assoc_legendrel( unsigned int n, unsigned int m, long double x );
   double      assoc_legendre( unsigned int n, unsigned int m, IntegralType x ); \fB(2)\fP

   1) Computes the associated Legendre polynomials of the degree n, order m, and
   argument x.
   2) A set of overloads or a function template accepting an argument of any integral
   type. Equivalent to \fB(1)\fP after casting the argument to double.

   As all special functions, assoc_legendre is only guaranteed to be available in
   <cmath> if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at
   least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before
   including any standard library headers.

.SH Parameters

   n - the degree of the polynomial, a value of unsigned integer type
   m - the order of the polynomial, a value of unsigned integer type
   x - the argument, a value of a floating-point or integral type

.SH Return value

   If no errors occur, value of the associated Legendre polynomial Pm
   n of x, that is (1 - x2
   )m/2

   dm
   dxm

   P
   n(x), is returned (where P
   n(x) is the unassociated Legendre polynomial, std::legendre(n, x)).

.SH Error handling

   Errors may be reported as specified in math_errhandling.

     * If the argument is NaN, NaN is returned and domain error is not reported.
     * If |x| > 1, a domain error may occur.
     * If n is greater or equal to 128, the behavior is implementation-defined.

.SH Notes

   Implementations that do not support TR 29124 but support TR 19768, provide this
   function in the header tr1/cmath and namespace std::tr1.

   An implementation of this function is also available in boost.math.

   The first few associated Legendre polynomials are:

     * assoc_legendre(0, 0, x) = 1.
     * assoc_legendre(1, 0, x) = x.
     * assoc_legendre(1, 1, x) = -(1 - x2
       )1/2
       .
     * assoc_legendre(2, 0, x) =

       1
       2

       (3x2
       - 1).
     * assoc_legendre(2, 1, x) = -3x(1 - x2
       )1/2
       .
     * assoc_legendre(2, 2, x) = 3(1 - x2
       ).

.SH Example

   (works as shown with gcc 6.0)


// Run this code

 #define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
 #include <cmath>
 #include <iostream>

 double P20(double x)
 {
     return 0.5 * (3 * x * x - 1);
 }

 double P21(double x)
 {
     return -3.0 * x * std::sqrt(1 - x * x);
 }

 double P22(double x)
 {
     return 3 * (1 - x * x);
 }

 int main()
 {
     // spot-checks
     std::cout << std::assoc_legendre(2, 0, 0.5) << '=' << P20(0.5) << '\\n'
               << std::assoc_legendre(2, 1, 0.5) << '=' << P21(0.5) << '\\n'
               << std::assoc_legendre(2, 2, 0.5) << '=' << P22(0.5) << '\\n';
 }

.SH Output:

 -0.125=-0.125
 -1.29904=-1.29904
 2.25=2.25

.SH See also

   legendre  Legendre polynomials
   legendref \fI(function)\fP
   legendrel

.SH External links

   Weisstein, Eric W. "Associated Legendre Polynomial." From MathWorld--A Wolfram Web
   Resource.
